Abstract
Let B n be the braid group on n ⩾ 4 strands. We prove that B n modulo its center is co-Hopfian. We then show that any injective endomorphism of B n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B n to B n + 1 is geometric for n ⩾ 7 . Additionally, we obtain analogous results for mapping class groups of punctured spheres. The methods use Thurston's theory of surface homeomorphisms and build upon work of Ivanov–McCarthy.
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